Thermal stitching: Combining the advantages of different quantum fermion solvers

For quantum fermion problems, many accurate solvers are limited by the temperature regime in which they can be usefully applied. The Mermin theorem implies the uniqueness of an effective potential from which both the exact density and free energy at a target temperature can be found, via a calculation at a different, reference temperature. We derive exact expressions for both the potential and the free energy in such a calculation and introduce three controllable approximations that reduce the cost of such calculations. We illustrate the effective potential and its free energy, and test the approximations, on the asymmetric two-site Hubbard model at finite temperature.

Figure 1

$n_1$ vs $x$ at $\tau =0.25$ (blue) and $\tau =1$ (red). Solid lines are $U=1$ and dashed are noninteracting, $U=0$. The intersections with the horizontal line at $n_1=0.5$ give the $v$ values that yield $n_1=0.5$ for the given temperature and interaction.

Figure 2

Effective thermal potential for $v=1$ and ${\tau }_R=1$ for various correlation strengths. Solid curves are interacting, dashed are noninteracting, and dot-dashed is Hxc. All calculations yield ${\overline{v}}_{\mathrm{Hxc},\tau }^{{\tau }_R}\le 0$.

Figure 3

Same as previous figure, but now for the difference between ETP and its reference.

Figure 4

Temperature dependence of the free energy and its deviation from reference for the same systems as in the previous figures.

Figure 5

Correlation free energy from ETP for the same system as previous figures. Solid is exact, dashed is from Eq. (16), and dot-dashed includes the further approximation of Eq. (17).